For the lower bound, the argument is essentially the same as in [1]}. Let us first give some intuition. If the distribution is supported on a finite number of points, e.g., two, there is an exponentially small probability that all the \(n\)  samples will contain the same point \((x,y)\) , so it will not help the learner distinguish between the functions \(h \in \mathcal {H}\)  that label this point correctly. For completeness, we present the argument formally in proof:exp-lower-bound.
